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arXiv:1712.08219

Duration of Classicality of Homogeneous Condensates with Attractive Interactions

Sankha Subhra Chakrabarty, Seishi Enomoto, Yaqi Han, P. Sikivie, Elisa Todarello

Published Dec 21, 2017• 0 citations
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Abstract

Dark matter axions and other highly degenerate bosonic fluids are commonly described by classical field equations. In a recent paper \cite{BECprop} we calculated the duration of classicality of homogeneous condensates with attractive contact interactions and of self-gravitating homogeneous condensates in critical expansion. According to their classical equations of motion, such condensates persist forever. In their quantum evolution parametric resonance causes quanta to jump in pairs out of the condensate into all modes with wavevector less than some critical value. We estimated in each case the time scale over which the condensate is depleted and after which a classical description is invalid.

DOI
10.48550/arxiv.1712.08219

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Dark matter axions and other highly degenerate bosonic fluids are commonly described by classical field equations. In a recent paper \cite{BECprop} we calculated the duration of classicality of homogeneous condensates with attractive contact interactions and of self-gravitating homogeneous condensates in critical expansion. According to their classical equations of motion, such condensates persist forever. In their quantum evolution parametric resonance causes quanta to jump in pairs out of the condensate into all modes with wavevector less than some critical value. We estimated in each case the time scale over which the condensate is depleted and after which a classical description is invalid.
Dark matter axions and other highly degenerate bosonic fluids are commonly described by classical field equations. In a recent paper \cite{BECprop} we calculated the duration of classicality of homogeneous condensates with attractive contact interactions and of self-gravitating homogeneous condensates in critical expansion. According to their classical equations of motion, such condensates persist forever. In their quantum evolution parametric resonance causes quanta to jump in pairs out of the condensate into all modes with wavevector less than some critical value. We estimated in each case the time scale over which the condensate is depleted and after which a classical description is invalid.
Quantum fields with large degeneracy are often approximated as classical fields. Here, we show how the quantum and classical evolution of a highly degenerate quantum field with repulsive contact self-interactions differ from each other. Initially, the field is taken to be homogeneous except for a small plane-wave perturbation in only one mode. In quantum field theory, modes satisfying both momentum and energy conservation of the quasiparticles, grow exponentially with time. However, in the classical field approximation, the system is stable. We calculate the time scale after which the classical field description becomes invalid.
We study the collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction. Equilibrium states in which the gravitational attraction and the attraction due to the self-interaction are counterbalanced by the quantum pressure (Heisenberg's uncertainty principle) exist only below a maximum mass ${M}_{\mathrm{max}}=1.012\ensuremath{\hbar}/\sqrt{Gm|{a}_{s}|}$ where as<0{a}_{s}<0as​<0 is the scattering length of the bosons and mmm is their mass [P. H. Chavanis, Phys. Rev. D 84, 043531 (2011)]. For M>MmaxM>{M}_{\mathrm{max}}M>Mmax​ the system is expected to collapse and form a black hole. We study the collapse dynamics by making a Gaussian ansatz for the wave function and reducing the problem to the study of the motion of a particle in an effective potential. We find that the collapse time scales as $(M/{M}_{\mathrm{max}}\ensuremath{-}1{)}^{\ensuremath{-}1/4}$ for $M\ensuremath{\rightarrow}{M}_{\mathrm{max}}^{+}$ and as ${M}^{\ensuremath{-}1/2}$ for $M\ensuremath{\gg}{M}_{\mathrm{max}}$. Other analytical results are given above and below the critical point corresponding to a saddle-node bifurcation. We apply our results to QCD axions with mass $m={10}^{\ensuremath{-}4}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}5.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}53}\text{ }\text{ }\mathrm{m}$ for which ${M}_{\mathrm{max}}=6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ and $R=3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{R}_{\ensuremath{\bigodot}}$. We confirm our previous claim that bosons with attractive self-interaction, such as QCD axions, may form low mass stars (axion stars or dark matter stars) but cannot form dark matter halos of relevant mass and size. These mini axion stars could be the constituents of dark matter. They can collapse into mini black holes of mass $\ensuremath{\sim}{10}^{\ensuremath{-}14}{M}_{\ensuremath{\bigodot}}$ in a few hours. In that case, dark matter halos would be made of mini black holes. We also apply our results to ultralight axions with mass $m=1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\text{ }\mathrm{eV}/{c}^{2}$ and scattering length ${a}_{s}=\ensuremath{-}8.29\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}60}\text{ }\text{ }\mathrm{fm}$ for which ${M}_{\mathrm{max}}=0.39\ifmmode\times\else\texttimes\fi{}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ and R=33  pcR=33\text{ }\text{ }\mathrm{pc}R=33  pc. These ultralight axions could cluster into dark matter halos. Axionic dark matter halos with attractive self-interaction can collapse into supermassive black holes of mass $\ensuremath{\sim}1{0}^{6}{M}_{\ensuremath{\bigodot}}$ (similar to those reported at the center of galaxies) in about one million years. We point out the limitations of the Gaussian ansatz to describe the late stages of the collapse dynamics. We also mention the possibility that, instead of forming a black hole, the collapse may be accompanied by a burst or relativistic axions (bosenova) leading to a cycle of collapses and explosions as observed for nongravitational Bose-Einstein condensates with attractive self-interaction.
In this paper, we study the Bose-Einstein condensation of a scalar field with an attractive self-interaction, with or without gravitational interactions. We confirm through full dynamical simulation that the condensation timescale due to self-interaction is inversely proportional to the square of the number density nnn and the self-coupling constant $g:\ensuremath{\tau}\ensuremath{\propto}{n}^{\ensuremath{-}2}{g}^{\ensuremath{-}2}$. We also investigate the condensation timescale when self-interaction and gravity are both important by solving the Gross-Pitaevskii-Poisson equations, and find that the condensation time scales according to an additive model for the cross section. We discuss the relevance of our results to theoretical models of boson star formation by condensation.
Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a "quantum phase transition" takes place involving 10 3 to 10 11 atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 10 13 /cm 3 , and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrödinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic "laser-speckle" type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.